However, simple contagion model can not deal with the spread of more complex social phenomenon, especially when the behaviors are risky, costly or polarized, ranging from the spread of public health behaviors (e.g., vaccine, vaping, diet) to social movement [1]}, [2]}, [3]}, [4]}. For instance, a well-known complex mechanism of social contagion is called social reinforcement, which represents the fact that exposure to multiple sources would give individuals more confidence to participate than multiple exposures to the same source [5]}, [6]}. Nevertheless, this empirically confirmed phenomenon is indistinguishable in the view of simple contagion model [7]}, [8]}, [9]}. Hence, more complex mechanisms are further incorporated into dynamical models, i.e., the complex contagion models, which lead to abundant intriguing phenomena, some of which even conflict with those simple contagion conclusions [10]}, [11]}, [12]}. For example, threshold models assumed that social contagion only happens when the number (or fraction) of exposures to multiple sources exceeds a given threshold and suggested the dependence of diffusion results on initial density of adopters [13]}. In addition, Centola et al. proved that the clustering network structure suppressed simple contagion while surprisingly facilitated the spread of behaviors that require social reinforcement [14]}. Another framework modified epidemic-like models by increasing transmission rate when exposing to multiple sources [15]}, [16]}. In particular, Iacopini et al. modeled social reinforcement from group discussions as higher-order interaction and showed the emergence of bistable region, where healthy and endemic states co-exist [17]}.
